Question: Simplify; express your answer in exponential form. Assume $t\neq 0, z\neq 0$. $\dfrac{{(t^{4}z^{5})^{4}}}{{(t^{-1}z^{-4})^{-5}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(t^{4}z^{5})^{4} = (t^{4})^{4}(z^{5})^{4}}$ On the left, we have ${t^{4}}$ to the exponent ${4}$ . Now ${4 \times 4 = 16}$ , so ${(t^{4})^{4} = t^{16}}$ Apply the ideas above to simplify the equation. $\dfrac{{(t^{4}z^{5})^{4}}}{{(t^{-1}z^{-4})^{-5}}} = \dfrac{{t^{16}z^{20}}}{{t^{5}z^{20}}}$ Break up the equation by variable and simplify. $\dfrac{{t^{16}z^{20}}}{{t^{5}z^{20}}} = \dfrac{{t^{16}}}{{t^{5}}} \cdot \dfrac{{z^{20}}}{{z^{20}}} = t^{{16} - {5}} \cdot z^{{20} - {20}} = t^{11}$